Lemma 43.22.2. Let X \to P be a closed immersion of nonsingular varieties. Let C' \subset P \times \mathbf{P}^1 be a closed subvariety of dimension r + 1. Assume
the fibre C = C'_0 has dimension r, i.e., C' \to \mathbf{P}^1 is dominant,
C' intersects X \times \mathbf{P}^1 properly,
[C]_ r intersects X properly.
Then setting \alpha = [C]_ r \cdot X viewed as cycle on X and \beta = C' \cdot X \times \mathbf{P}^1 viewed as cycle on X \times \mathbf{P}^1, we have
\alpha = \text{pr}_{X, *}(\beta \cdot X \times 0)
as cycles on X where \text{pr}_ X : X \times \mathbf{P}^1 \to X is the projection.
Proof.
Let \text{pr} : P \times \mathbf{P}^1 \to P be the projection. Since we are proving an equality of cycles it suffices to think of \alpha , resp. \beta as a cycle on P, resp. P \times \mathbf{P}^1 and prove the result for pushing forward by \text{pr}. Because \text{pr}^*X = X \times \mathbf{P}^1 and \text{pr} defines an isomorphism of C'_0 onto C the projection formula (Lemma 43.22.1) gives
\text{pr}_*([C'_0]_ r \cdot X \times \mathbf{P}^1) = [C]_ r \cdot X = \alpha
On the other hand, we have [C'_0]_ r = C' \cdot P \times 0 as cycles on P \times \mathbf{P}^1 by Lemma 43.17.1. Hence
[C'_0]_ r \cdot X \times \mathbf{P}^1 = (C' \cdot P \times 0) \cdot X \times \mathbf{P}^1 = (C' \cdot X \times \mathbf{P}^1) \cdot P \times 0
by associativity (Lemma 43.20.1) and commutativity of the intersection product. It remains to show that the intersection product of C' \cdot X \times \mathbf{P}^1 with P \times 0 on P \times \mathbf{P}^1 is equal (as a cycle) to the intersection product of \beta with X \times 0 on X \times \mathbf{P}^1. Write C' \cdot X \times \mathbf{P}^1 = \sum n_ k[E_ k] and hence \beta = \sum n_ k[E_ k] for some subvarieties E_ k \subset X \times \mathbf{P}^1 \subset P \times \mathbf{P}^1. Then both intersections are equal to \sum m_ k[E_{k, 0}] by Lemma 43.17.1 applied twice. This finishes the proof.
\square
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